The Module of Logarithmic p-Forms of a Locally Free Arrangement

Abstract

For an essential, central hyperplane arrangement A⊆V≃kn+1 we show that Ω1(A) (the module of logarithmic one forms with poles along A) gives rise to a locally free sheaf on Pn if and only if, for all X∈LA with rank X<dimV, the module Ω1(AX) is free. Motivated by a result of L. Solomon and H. Terao (1987, Adv. Math.64, 305–325), we give a formula for the Chern polynomial of a bundle E on Pn in terms of the Hilbert series of ⊕m∈ZH0(Pn,∧iE(m)). As a corollary, we prove that if the sheaf associated to Ω1(A) is locally free, then π(A,t) is essentially the Chern polynomial. If Ω1(A) has projective dimension one and is locally free, we give a minimal free resolution for Ωp and show that ΛpΩ1(A)≃Ωp(A), generalizing results of L. Rose and H. Terao (1991, J. Algebra136, 376–400) on generic arrangements.